Version 1, wide-open. Here I covered the dollar bill with 45 pennies, going for a hexagonal lattice. I'm guessing this covered a percentage of the dollar in the mid- to high-80's. (One easily shows that an infinite hexagonal lattice covers a fraction Pi*Sqrt[3]/6, or a little less than 91%, so I'm assuming the below is pushing up against that limit.)

Version 2, intermediate. I nudged the configuration above in order to make room for the additional coins to touch the dollar bill - that seemed to force a decision to remove five of the pennies and replace the nickel with a dime.

I did some quickie image analysis on the above awful picture to give me some inputs for a Monte Carlo calculation of the covered area:

The result was just under 85% coverage. I didn't do a sensitivity analysis. I'm guessing this configuration is low- to mid-80's.

Version 3, tight. Here, my attempt uses 8 dimes, 15 pennies and a nickel. According to the spreadsheet, this configuration covers just over 64% of the dollar.

It would be nice to be able to replace that nickel with 5 pennies - according to the spreadsheet, that would bump the area up to 74.6%. But I wasn't seeing quite enough room to do that, and truth be told I have put a lot more time into blogging this puzzle than I've spent trying to solve it. Feel free to blow these attempts out of the water!

***

I thought of this puzzle after daydreaming the following exchange between a bank teller and an eccentric customer:

"May I help you?"
"I'd like change for a dollar."
"Here you go!" (Hands the customer four quarters.)
"Why, you cheat! I had 16 square inches' worth of currency and you gave me back only 3 square inches' worth!"
"I'm sorry, sir. Here are 7 dimes and 30 pennies instead."
"OK, close enough...."

I thought it was amusing, this image of a person valuing an exchange along a totally irrelevant dimension. As if one were to share a long kiss with somebody, and then become upset because the other person took a different number of breaths than you did.

***

The above puzzle takes the form, "How many X can you cover with Y?" I'm sure there a lot of problems and solutions of this nature to be found in the literature on circle-packing. I don't know that literature, but a quick search just now turned up some neat pictures here.

3 comments:

Enjoyed reading about "Cents on the Dollar"! I don't have enough change on me to try it out, but from the final picture it looks like you could shift the three right dimes to the lower left corner, remove the nickel, and then add five pennies. It would be close anyway.

Hope all is well with you all! Natalia and I have moved to Virginia; she got a job at Washington and Lee, and I'm now at Roanoke College. We were really lucky that those two jobs came along in the same season. If you're ever in SW Virginia (or just want to come here), let us know.

The reason I actually thought of you is that a friend sent me a link to a text-adventure game whose style reminds me of the hilarious "Bigfoot" book you gave us a few years ago. I don't know if you enjoy text-adventures, but here's the link: http://www.grunk.org/lostpig/

Jason Zimba was a lead writer of the Common Core State Standards for Mathematics and is a Founding Partner of Student Achievement Partners, a non-profit organization. He holds a B.A. from Williams College with a double major in mathematics and astrophysics; an M.Sc. by research in mathematics from the University of Oxford; and a Ph.D. in mathematical physics from the University of California at Berkeley. As a researcher, Dr. Zimba’s work spanned a range of fields, including astronomy, astrophysics theoretical physics, philosophy of science, and pure mathematics. His academic awards include a Rhodes scholarship and a Majorana Prize for theoretical physics. Dr. Zimba has taught physics and mathematics to university students and high school students, as well as to adult prison inmates and members of other disadvantaged groups. He is the author of Force and Motion: An Illustrated Guide to Newton’s Laws.